What does it take to prove that a quadrilateral is a rhombus?

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Parallelogram Rhombus Rectangle Square Trapezoid Isosceles Trapezoid
Exactly 1 pair of parallel sides
2 pairs of parallel sides
Exactly 1 pair of congruent sides
2 pairs of congruent sides
All sides congruent
Perpendicular diagonals
Congruent diagonals
Diagonals bisect angles
Diagonals bisect each other
Opposite angles congruent
Four right angles
Base angles congruent
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Parallelogram Rhombus Rectangle Square Trapezoid Isosceles Trapezoid
Exactly 1 pair of parallel sides
2 pairs of parallel sides
Exactly 1 pair of congruent sides
2 pairs of congruent sides
All sides congruent
Perpendicular diagonals
Congruent diagonals
Diagonals bisect angles
Diagonals bisect each other
Opposite angles congruent
Four right angles
Base angles congruent
What does it take to prove that a quadrilateral
is a rhombus?
We know a rhombus has perpendicular diagonals.
Does this mean that if we have a
quadrilateral with perpendicular diagonals, it
must be a rhombus?
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Ways to prove that a quadrilateral is a
parallelogram
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Ways to prove that a quadrilateral is a
parallelogram Show both pairs of opposite sides
parallel
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Ways to prove that a quadrilateral is a
parallelogram Show both pairs of opposite sides
parallel Show one pair of opposite sides parallel
and congruent  
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Ways to prove that a quadrilateral is a
parallelogram Show both pairs of opposite sides
parallel Show one pair of opposite sides parallel
and congruent Show both pairs of opposite sides
congruent Show both pairs of opposite angles
congruent Show that the diagonals bisect each
other  
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Ways to prove that a quadrilateral is a
parallelogram Show both pairs of opposite sides
parallel Show one pair of opposite sides parallel
and congruent Show both pairs of opposite sides
congruent Show both pairs of opposite angles
congruent Show that the diagonals bisect each
other   Ways to prove that a quadrilateral is a
rhombus parallelogram and two adjacent sides
congruent
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Ways to prove that a quadrilateral is a
parallelogram Show both pairs of opposite sides
parallel Show one pair of opposite sides parallel
and congruent Show both pairs of opposite sides
congruent Show both pairs of opposite angles
congruent Show that the diagonals bisect each
other   Ways to prove that a quadrilateral is a
rhombus parallelogram and two adjacent sides
congruent parallelogram and perpendicular
diagonals
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Ways to prove that a quadrilateral is a
parallelogram Show both pairs of opposite sides
parallel Show one pair of opposite sides parallel
and congruent Show both pairs of opposite sides
congruent Show both pairs of opposite angles
congruent Show that the diagonals bisect each
other   Ways to prove that a quadrilateral is a
rhombus parallelogram and two adjacent sides
congruent parallelogram and perpendicular
diagonals parallelogram and diagonals bisect its
angles  
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Ways to prove that a quadrilateral is a
parallelogram Show both pairs of opposite sides
parallel Show one pair of opposite sides parallel
and congruent Show both pairs of opposite sides
congruent Show both pairs of opposite angles
congruent Show that the diagonals bisect each
other   Ways to prove that a quadrilateral is a
rhombus parallelogram and two adjacent sides
congruent parallelogram and perpendicular
diagonals parallelogram and diagonals bisect its
angles   Ways to prove that a quadrilateral is a
rectangle parallelogram and one right
angle parallelogram and congruent
diagonals   Ways to prove that a quadrilateral is
an isosceles trapezoid trapezoid and
non-parallel sides congruent trapezoid and one
pair of congruent base angles trapezoid and
congruent diagonals
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The median of a triangle is a segment from a
vertex to the midpoint of the opposite side.
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(No Transcript)
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.
½x
x
The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
½ (b1 b2)
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.

?
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.

?
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.

?
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.

?
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.

?
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.

?
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The segment joining the midpoints of two sides of
a triangle (the midsegment) is parallel to the
third side and half its length.
?
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
Where is the flaw in this proof?
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
First we prove the following lemma If two lines
are parallel to the same line, they are parallel
to each other.
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
First we prove the following lemma If two lines
are parallel to the same line, they are parallel
to each other.
If m // l and n // l, prove m // n
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
First we prove the following lemma If two lines
are parallel to the same line, they are parallel
to each other.
If m // l and n // l, prove m // n
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
It is a midsegment of ?ABD
Definition of trapezoid
It is a midsegment of ?BDC
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The median of a trapezoid is parallel to the
bases and is equal in length to the average of
the lengths of the bases.
Proof
½ b2
½ b1
MN ½ b1 ½ b2 ½ (b1 b2)
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29 inches
42 inches
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The altitude of a parallelogram
The altitude of a trapezoid
C
D
B
A
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The following construction problems should be
worked on and solved in groups of three people.
Each group will be assessed three timesonce for
each category. When your group is ready to be
assessed, notify me and indicate on which
category the group wants to be assessed. Your
group will choose the problem in that category on
which to be assessed. The group member who will
be asked to solve the problem will be selected
randomly. Whatever grade is assigned to that
student will be the grade for the group. Be sure
each group member can solve each problem
correctly. The following scoring guidelines will
be used to assess each problem. The maximum
possible point total is 45 points.
Construction Construction
0 Cannot do the construction correctly even with hints
3 Can do the construction correctly but only after hints are given
6 Can do the construction correctly without hints
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The following construction problems should be
worked on and solved in groups of three people.
Each group will be assessed three timesonce for
each category. When your group is ready to be
assessed, notify me and indicate on which
category the group wants to be assessed. Your
group will choose the problem in that category on
which to be assessed. The group member who will
be asked to solve the problem will be selected
randomly. Whatever grade is assigned to that
student will be the grade for the group. Be sure
each group member can solve each problem
correctly. The following scoring guidelines will
be used to assess each problem. The maximum
possible point total is 45 points.
Proof Proof
0 Cannot do the proof correctly even with hints
3 Can do the proof correctly but only after hints are given
6 Can do the proof correctly without hints
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Sample Construct a rhombus given one side and
an altitude 1. Construct a rhombus (not a
square), given a. one side and one angle
(1 point) b. one angle and a diagonal (2
points) c. the altitude and one diagonal (3
points) 2. Construct a parallelogram (not a
rhombus), given a. one side, one angle, and one
diagonal (1 point) b. two adjacent sides and
an altitude (2 points) c. one angle, one
side, and the altitude on that side (3 points)
3. Construct an isosceles trapezoid, given
a. the diagonal, altitude, and one of the bases
(1 point) b. one base, the diagonal, and
the angle included by them (2 points) c.
the bases and one angle (3 points)
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
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Construct a rhombus given one side and an
altitude.
Conclusion ARTB is a rhombus with
the required information.
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Construct a rhombus given one side and an
altitude.
Conclusion ARTB is a rhombus with
the required
information. Brief Proof Because AR, AB, and
RT, are all radii of congruent circles, they have
the same length. Because RT and AB are both
perpendicular to PQ, RT is parallel to AB.
Therefore, ARTB a parallelogram (one pair of
opposite sides are both congruent and parallel).
It is now also a rhombus because two adjacent
sides are congruent (AR ? AB). The sides of the
rhombus are all congruent to AB, which was the
given side. Since QT is parallel to AB, the
distance from QT to AB is always the same, and
that distance is equal to PQ, the given altitude.